Subgroups — #26123540

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689107	C₂	1	2	yes	maximal
3	C₂×#15688888	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₇⋊#338	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#408	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093960	—	2	4	no	—
9	C₂×#13093960	—	2	2	no	—
10	#15689107	—	2	4	no	—
11	#15689107	—	2	4	no	—
12	C₂×#13093943	—	2	2	no	—
13	C₂×#13093941	—	2	2	no	—
14	#15688888	—	2	2	no	—
15	C₄₇⋊#338	—	2	4	no	—
16	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#408	—	2	2	no	—
20	C₄₇×#408	—	2	2	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₄	1	4	yes	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	—	2	2	no	—
32	C₂×#10625329	D₄	1	4	yes	—
33	C₂×#10625329	D₄	1	4	yes	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	#13093960	—	2	8	no	—
37	#13093943	—	2	8	no	—
38	#13093941	—	2	8	no	—
39	C₂×#41×C₄₇	—	2	2	no	—
40	C₂×#41×C₄₇	—	2	2	no	—
41	C₂×#41×C₄₇	—	2	1	no	—
42	C₂×#41×C₄₇	—	2	1	no	—
43	C₂×#41×C₄₇	—	2	1	no	—
44	C₂×#41×C₄₇	—	2	1	no	—
45	C₂×#41×C₄₇	—	2	2	no	—
46	C₂×#41×C₄₇	—	2	2	no	—
47	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₄₇	Q₈	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₄₇	—	2	2	no	—
53	C₂×M₁₆×C₄₇	—	2	1	no	—
54	C₂×M₁₆×C₄₇	—	2	1	no	—
55	C₂×M₁₆×C₄₇	—	2	2	no	—
56	C₂²×C₈×C₄₇	—	2	2	no	—
57	C₂²×C₈×C₄₇	—	2	2	no	—
58	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₄₇	D₄	1	2	yes	—
61	C₂²×C₈×C₄₇	D₄	1	2	yes	—
62	C₂²⋊C₈×C₄₇	—	2	8	no	—
63	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	#41×C₄₇	—	4	1	no	—
66	#41×C₄₇	—	4	2	no	—
67	#41×C₄₇	—	4	2	no	—
68	#41×C₄₇	—	4	2	no	—
69	#41×C₄₇	—	4	1	no	—
70	#41×C₄₇	—	4	2	no	—
71	#41×C₄₇	—	4	2	no	—
72	#41×C₄₇	—	4	1	no	—
73	#41×C₄₇	—	4	2	no	—
74	#41×C₄₇	—	4	1	no	—
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂×Q₈×C₄₇	—	2	1	no	—
77	C₂×Q₈×C₄₇	—	2	1	no	—
78	C₂×Q₈×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	2	4	no	—
82	C₂×D₄×C₄₇	—	2	4	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	2	1	no	—
85	C₂×D₄×C₄₇	—	2	1	no	—
86	C₂×D₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	—	2	2	no	—
89	C₂²×C₄×C₄₇	—	2	4	no	—
90	C₂²×C₄×C₄₇	—	2	4	no	—
91	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
92	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
99	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
100	C₂²×C₄×C₄₇	#41	1	2	yes	—
101	C₂²×C₄×C₄₇	#41	1	2	yes	—
102	M₁₆×C₄₇	—	4	2	no	—
103	M₁₆×C₄₇	—	4	1	no	—
104	M₁₆×C₄₇	—	4	1	no	—
105	C₂×C₈×C₄₇	—	2	4	no	—
106	C₂×C₈×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	—	2	4	no	—
108	C₂×C₈×C₄₇	—	2	2	no	—
109	C₂×C₈×C₄₇	—	2	2	no	—
110	C₂×C₈×C₄₇	—	2	4	no	—
111	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
112	#10625329	—	2	4	no	—
113	#10625329	—	2	4	no	—
114	#10625329	—	2	4	no	—
115	#10625329	—	2	4	no	—
116	#10625329	—	4	2	no	—
117	#10625329	—	4	2	no	—
118	#10625329	—	4	2	no	—
119	#10625329	—	4	2	no	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	—	2	4	no	—
122	C₂×C₄₇⋊C₈	—	4	2	no	—
123	C₂×C₄₇⋊C₈	—	4	2	no	—
124	C₂×C₄₇⋊C₈	—	4	2	no	—
125	C₂×C₄₇⋊C₈	—	4	2	no	—
126	C₂³×C₄₇	—	2	2	no	—
127	C₂³×C₄₇	—	2	4	no	—
128	C₂³×C₄₇	—	4	2	no	—
129	C₂³×C₄₇	—	4	4	no	—
130	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
132	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
133	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
135	Q₈×C₄₇	—	4	1	no	—
136	Q₈×C₄₇	—	4	2	no	—
137	Q₈×C₄₇	—	4	1	no	—
138	D₄×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	4	4	no	—
140	D₄×C₄₇	—	4	2	no	—
141	D₄×C₄₇	—	4	1	no	—
142	D₄×C₄₇	—	4	2	no	—
143	D₄×C₄₇	—	4	1	no	—
144	C₂×C₄×C₄₇	—	2	2	no	—
145	C₂×C₄×C₄₇	—	2	4	no	—
146	C₂×C₄×C₄₇	—	4	2	no	—
147	C₂×C₄×C₄₇	—	4	4	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
154	C₂×C₄×C₄₇	#120	1	2	yes	—
155	C₂×C₄×C₄₇	#121	1	2	yes	—
156	C₂×C₄×C₄₇	#121	1	2	yes	—
157	C₂×C₄×C₄₇	#121	1	2	yes	—
158	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
161	C₂×C₄×C₄₇	#123	1	2	yes	—
162	C₂×C₄×C₄₇	#123	1	2	yes	—
163	C₂×C₄×C₄₇	#124	1	2	yes	—
164	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
165	C₂×C₄×C₄₇	#126	1	2	yes	—
166	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
167	C₈×C₄₇	—	2	4	no	—
168	C₈×C₄₇	—	4	2	no	—
169	C₄₇⋊C₈	—	4	4	no	—
170	C₄₇⋊C₈	—	4	4	no	—
171	#11099	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	2	no	—
173	C₂²×C₄₇	—	2	2	no	—
174	C₂²×C₄₇	—	2	2	no	—
175	C₂²×C₄₇	—	4	2	no	—
176	C₂²×C₄₇	—	4	2	no	—
177	C₂²×C₄₇	—	4	4	no	—
178	C₂²×C₄₇	—	4	2	no	—
179	C₂²×C₄₇	—	8	4	no	—
180	C₂²×C₄₇	#380	1	1	yes	characteristic
181	C₂²×C₄₇	#385	1	2	yes	—
182	C₂²×C₄₇	#393	1	1	yes	characteristic
183	C₂²×C₄₇	#396	1	2	yes	—
184	C₂²×C₄₇	#399	1	1	yes	characteristic
185	C₄×C₄₇	—	4	2	no	—
186	C₄×C₄₇	#386	1	1	yes	characteristic
187	C₄×C₄₇	#388	1	2	yes	—
188	C₄×C₄₇	#390	1	1	yes	characteristic
189	C₄×C₄₇	#392	1	1	yes	characteristic
190	C₄×C₄₇	#394	1	2	yes	—
191	C₄×C₄₇	#397	1	1	yes	characteristic
192	C₂²×#108	—	47	1	no	—
193	C₂×#413	—	47	2	no	—
194	C₂×#408	D₄₇	1	1	yes	characteristic
195	C₂×#341	—	47	1	no	—
196	C₂×#338	—	47	2	no	—
197	C₂×C₄₇	—	2	2	no	—
198	C₂×C₄₇	—	2	1	no	—
199	C₂×C₄₇	—	2	2	no	—
200	C₂×C₄₇	—	2	1	no	—
201	C₂×C₄₇	—	8	2	no	—
202	C₂×C₄₇	#1439	1	1	yes	characteristic
203	C₂×C₄₇	#1813	1	2	yes	—
204	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
205	C₂²×M₁₆	—	47	2	no	—
206	C₂²×M₁₆	—	47	2	no	—
207	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
208	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
209	C₂×#108	—	94	4	no	—
210	C₂×#108	—	94	2	no	—
211	#413	—	94	4	no	—
212	#413	—	94	4	no	—
213	C₂×#101	—	94	2	no	—
214	C₂×#100	—	94	2	no	—
215	#408	—	2	2	no	—
216	#408	—	2	2	no	—
217	C₂×C₂²⋊C₈	Dic₄₇	1	4	yes	—
218	#341	—	94	2	no	—
219	#338	—	94	4	no	—
220	C₄₇	#11099	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	2	1	no	—
222	C₂×#41	—	2	1	no	—
223	C₂×#41	—	2	2	no	—
224	C₂×#41	—	2	2	no	—
225	C₂×#41	—	2	2	no	—
226	C₂×#41	—	2	2	no	—
227	C₂×#41	—	2	1	no	—
228	C₂×#41	—	2	1	no	—
229	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
230	C₂²×D₄	Dic₉₄	1	2	yes	—
231	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
232	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
233	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	47	4	no	—
239	C₂×M₁₆	—	47	4	no	—
240	C₂×M₁₆	—	94	2	no	—
241	C₂×M₁₆	—	94	2	no	—
242	C₂×M₁₆	—	94	2	no	—
243	C₂×M₁₆	—	94	2	no	—
244	C₂×M₁₆	—	94	2	no	—
245	C₂×M₁₆	—	94	2	no	—
246	C₂×M₁₆	—	94	2	no	—
247	C₂×M₁₆	—	94	2	no	—
248	C₂×M₁₆	—	94	2	no	—
249	C₂×M₁₆	—	94	2	no	—
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	—	2	2	no	—
252	C₂²×C₈	—	94	2	no	—
253	C₂²×C₈	—	94	2	no	—
254	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
255	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
256	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
257	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
258	#108	—	94	8	no	—
259	#101	—	94	8	no	—
260	#100	—	94	8	no	—
261	C₂²⋊C₈	—	2	8	no	—
262	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
263	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
264	#41	—	4	1	no	—
265	#41	—	4	2	no	—
266	#41	—	4	2	no	—
267	#41	—	4	2	no	—
268	#41	—	4	1	no	—
269	#41	—	4	2	no	—
270	#41	—	4	2	no	—
271	#41	—	4	1	no	—
272	#41	—	4	2	no	—
273	#41	—	4	1	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×D₄	—	2	4	no	—
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
291	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
292	C₂²×C₄	#10625350	1	1	yes	characteristic
293	C₂²×C₄	#10625350	1	1	yes	characteristic
294	C₂²×C₄	D₄×D₄₇	1	2	yes	—
295	C₂²×C₄	D₄×D₄₇	1	2	yes	—
296	C₂²×C₄	#10625352	1	2	yes	—
297	C₂²×C₄	#10625352	1	2	yes	—
298	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
301	M₁₆	—	4	2	no	—
302	M₁₆	—	4	1	no	—
303	M₁₆	—	4	1	no	—
304	M₁₆	—	94	4	no	—
305	M₁₆	—	94	4	no	—
306	M₁₆	—	94	4	no	—
307	M₁₆	—	94	4	no	—
308	M₁₆	—	188	2	no	—
309	M₁₆	—	188	2	no	—
310	M₁₆	—	188	2	no	—
311	M₁₆	—	188	2	no	—
312	C₂×C₈	—	2	4	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	94	4	no	—
319	C₂×C₈	—	94	4	no	—
320	C₂×C₈	—	188	2	no	—
321	C₂×C₈	—	188	2	no	—
322	C₂×C₈	—	188	2	no	—
323	C₂×C₈	—	188	2	no	—
324	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	4	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	4	no	—
329	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
330	C₂³	#13094001	1	2	yes	—
331	C₂³	#13094001	1	2	yes	—
332	C₂³	#13094003	1	1	yes	characteristic
333	C₂³	#13094003	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	2	no	—
336	Q₈	—	4	1	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	1	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
349	C₂×C₄	#13093983	1	1	yes	characteristic
350	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
351	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
352	C₂×C₄	#13093992	1	1	yes	characteristic
353	C₂×C₄	#13093994	1	2	yes	—
354	C₂×C₄	#13093995	1	2	yes	—
355	C₂×C₄	#13093996	1	2	yes	—
356	C₂×C₄	#13093997	1	2	yes	—
357	C₂×C₄	#13093998	1	1	yes	characteristic
358	C₂×C₄	#13093998	1	1	yes	characteristic
359	C₂×C₄	#13094005	1	1	yes	characteristic
360	C₂×C₄	#13094006	1	2	yes	—
361	C₂×C₄	#13094008	1	2	yes	—
362	C₂×C₄	#13094010	1	2	yes	—
363	C₂×C₄	#13094011	1	1	yes	characteristic
364	C₂×C₄	#13094017	1	2	yes	—
365	C₂×C₄	#13094020	1	1	yes	characteristic
366	C₈	—	2	4	no	—
367	C₈	—	4	2	no	—
368	C₈	—	188	4	no	—
369	C₈	—	188	4	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	8	4	no	—
378	C₂²	#15688983	1	1	yes	characteristic
379	C₂²	#15688998	1	2	yes	—
380	C₂²	#15689035	1	1	yes	characteristic
381	C₂²	#15689046	1	2	yes	—
382	C₂²	#15689059	1	1	yes	characteristic
383	C₄	—	4	2	no	—
384	C₄	#15689003	1	1	yes	characteristic
385	C₄	#15689016	1	2	yes	—
386	C₄	#15689026	1	1	yes	characteristic
387	C₄	#15689032	1	1	yes	characteristic
388	C₄	#15689039	1	2	yes	—
389	C₄	#15689048	1	1	yes	characteristic
390	C₂	—	2	2	no	—
391	C₂	—	2	1	no	—
392	C₂	—	2	2	no	—
393	C₂	—	2	1	no	—
394	C₂	—	8	2	no	—
395	C₂	#20241633	1	1	yes	characteristic
396	C₂	#20243438	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#408	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#408	D₄₇
hypocenter	C₄₇	#11099
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#408

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#13093960	C₂	semidirect
3. C₂×#15688888	C₂	semidirect
4. C₂×C₄₇⋊#338	C₂	semidirect
220. C₄₇	#11099	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13093960	2
C₂×#15689107	2
C₂×#15688888	2
C₂×C₄₇⋊#338	2
C₂×C₄₇×#408	2
#11099	47